Accurate approximation of stochastic differential equations Simon J.A. Malham and Anke Wiese (Heriot Watt University, Edinburgh) Birmingham: 6th February 29
Stochastic differential equations dy t = V (y t )dt + V 1 (y t )dwt 1 + + V d (y t )dwt d Four approaches to approximation, Solve: related PDE for a weak approximation (Monte Carlo) for a strong approximation pathwise First three: expectation and higher moments of solution sought.
Basic setting t y t = y + V (y τ )dτ + d i=1 t V i (y τ )dw i τ Stratonovich form (familiar, easier) Non-commuting vector fields: V i = n j=1 V j i (y) y j d-dimensional driving signal: (W 1,...,W d ) Convention: W t t Solution process: y : R + R N,
Wiener process 1.2 1.8.6.4.2 t.2 h 2h. (N 1) h T=Nh Q t t W t W s N(, t s) Independent increments Continuous, potentially nowhere differentiable
Applications Finance: Heston model for pricing options. The stock price is modelled as a stochastic process u with stochastic volatility v. in Itô form: du t = µu t dt + v t u t dw 1 t, dv t = α(θ v t )dt + βρ v t dw 1 t + β 1 ρ 2 v t dw 2 t, Neuronal dynamics (Coombes & Lord) Molecular DNA damage dynamics (Chickarmane et al. ) Chemical reactions (K. Burrage) Ocean/weather modelling Linear-quadratic optimal stochastic control.
Weak approximation Replace Gaussian increments W i (t n, t n+1 ) by simpler RVs Ŵ i (t n, t n+1 ) with appropriate moment properties, eg. by branching process: P ( Ŵ i (t n, t n+1 ) = ± h ) = 1 2 Expectation of approximation ŷ T across all paths at the final time T is close to the expectation of the true solution: E(y T ) E(ŷ T ) = O(h p ) No pathwise comparison: these paths not close to Wiener paths
Strong approximation (case hereafter) Generate approximate Wiener process paths Pick the increments W i (t n, t n+1 ) independently from N(, h) Since we have followed Wiener path approximations, we expect to be able to compare y T with ŷ T ; they re close in the sense: E y T ŷ T = O ( h p 2 )
Stochastic chain rule (Stratonovich) y t = y + i t V i y τ dw i τ Itô lemma (stochastic chain rule) f y t = f y + j t V j f y τ dw j τ E.g. choose f = V i V i y t = V i y + j t V j V i y τ dw j τ y t = y + i t V i y + j τ1 V j V i y τ2 dwτ j 2 dwτ i 1
Stochastic Taylor series y t = y + i t dw i τ 1 V i y + i,j t τ1 V j V i y τ2 dw j τ 2 dw i τ 1 Now choose f = V j V i y t = y + i t dwτ i 1 V i y + }{{} i,j J i (t) t τ1 dwτ j 2 dwτ i 1 V j V i y + } {{ } J ji (t) Feynman Dyson path ordered exponential, Neumann series, Peano Baker series, Chen-Fleiss series, stochastic B-series,... Euler-Maruyama and Milstein methods, RK methods (Kloeden & Platen) Need approximations for iterated integrals: quadrature later
Flow map y t = ϕ t y ϕ t = id + i J i V i + i,j J ji V j V i + i,j,k J kji V k V j V i + ϕ t = id + w A + J w V w Here A + = { non-empty words overa = {, 1,...,d} } Remainders and local error
Stochastic integral properties Expectations: E(J i ) = E(J ii ) = E( 1 2 J2 i ) = 1 2 h E(J ij ) =, i j Shuffle relations: J u J v = w sh(u,v) J w J a1 J a2 = J a1 a 2 + J a2 a 1 J a1 J a2 a 3 = J a1 a 2 a 3 + J a2 a 1 a 3 + J a2 a 3 a 1
Exponential Lie series Set ϕ t = expψ t then ψ t = lnϕ t = (ϕ t id) 1 2 (ϕ t id) 2 + 1 3 (ϕ t id) 3 + d = J i V i + 1 2 (J ij J ji )[V i, V j ] + i= i>j Local error: R ls = expψ t exp ˆψ t = ψ t ˆψ t + o(ψ t ˆψ t ) Magnus 1954, Chen 1957, Kunita 198, Strichartz 1987, Ben Arous 1989, Castell 1993, Castell Gaines 1995, Burrage 1999, Misawa 21, P-C. Moan 24.
Castell Gaines (ODE) method Truncated exponential Lie series across [t n, t n+1 ]: ˆψ tn,t n+1 = Ĵ 1 V 1 + Ĵ 2 V 2 + Ĵ V + 1 2 (Ĵ 12 Ĵ 21 )[V 1, V 2 ]. Approximate solution: y tn+1 exp( ˆψ tn,t n+1 ) y tn. Castell Gaines: solve ODE u (τ) = ˆψ tn,t n+1 u(τ) across τ [, 1] with u() = y tn gives u(1) y tn+1.
Quadrature How do we strongly approximate J 12 (t n, t n+1 )? By its conditional expection Ĵ 12 (t n, t n+1 ) Karhunen Loeve (Fourier) expansion Wiktorsson s method looks at the joint probability distribution function for J 1, J 2 and J 12. Error: Rough paths (Lyons) J 12 (t n, t n+1 ) Ĵ 12 (t n, t n+1 ) = h/ Q L2 Wiktorsson improves to h/q (SDELab: Gilsing & Shardlow)
Geometric stochastic integration M is a smooth submanifold of R n Lie group G with corresponding Lie algebra g Lie group action Λ y : G M; starting point y M fixed Λ transitive, effective Vector fields V i, i =, 1,...,d, are each infinitesimal Lie group actions generated by some ξ i g via Λ y, i.e. (Fundamental vector fields.) V i = λ ξi
Homogeneous manifolds o σ t σ^ t g log exp id St ^ St G Λ 1 y Λ y y y t ^y t M
Example: Stiefel manifold M = V n,k {y R n k : y T y = I } 1. G = SO(n) g = so(n) 2. Λ y S S y Direct calculation λ ξ y = ( ) (Λ y ) X ξ y = ξ(y)y Note S 2 = V 3,1.
Stochastic Munthe-Kaas methods Given smooth map ξ: M g. v ξ σ = dexp 1 σ ξ ( Λ y exp σ ) X ξ S = ξ ( Λ y S ) S λ ξ = ( Λ y ) X ξ v ξi exp Xξi (Λ y ) λ ξi V i λ ξi equivalent to original SDE
Rigid body (satellite): simulation
Rigid body: S 2 adherence 2 4 log(distance from manifold) 6 8 1 12 Stochastic Taylor Castell Gaines Munthe Kaas 14 16 1 2 3 4 5 6 7 8 9 1 time
Basic idea ϕ = id + w A + J w V w Suppose: ψ = F(ϕ) = C k (ϕ id) k = k=1 w A + K w V w where w C k K w = k=1 u 1,...,u k A + u 1 u 2 u k =w Goal: ˆϕ = F 1 ( ˆψ); choose best C k. J u1 J u2 J uk (t)
Hopf algebraic structure ϕ = 1 1 + ψ = k 1 w A + w w C k ( ϕ 1 1 ) k ) k = ( C k w w k 1 w A + = ( ) C k (u 1... u k ) (u 1...u k ) k 1 u 1,...,u k A + = ( w ) C k u 1... u k w w A = k=1 u 1,...,u k A + w=u 1...u k w A (K w) w.
Sinh-log expansion w C k K w = k=1 u 1,...,u k A + w=u 1...u k u 1 u 2... u k u 1 u 2... u k c u 1 1 sc u 2 1 s...sc u k 1 Sinh-log series coefficients K = 1 2( c n + (c s) n).
Future directions Numerical stability (Buckwar et al. ) Positivity preservation (Andersen,...) Implicit methods? (Kahl, Alfonsi,...) Symplectic methods (Tretyakov, Bou Rabee) Driving fractional Brownian motions (Baudoin,...) Driving processes with jumps, eg. Lévy processes PSDEs (Brown report)
Introductory references Theory: L.C. Evans: An introduction to stochastic differential equations http://math.berkeley.edu/ evans Numerics: D.J. Higham: An algorithmic introduction to numerical simulation of stochastic differential equations SIAM Review 43 (21), pp. 525 546
Brown report Applied mathematics at the US Department of Energy: from Sections 1 & 2: Develop new approaches for efficient modeling of large stochastic systems. Significantly advance the theory and tools for quantifying the effects of uncertainty and numerical simulation error on predictions using complex models and when fitting complex models to observations.
Itô s lemma dy t = Ṽ y t dt + d i=1 V i y t dw i t df (y t ) = Ṽ f y t dt+ d i=1 V i f y t dw i t + d i=1 V 2 i f y t dt df (y t ) = ( Ṽ + d i=1 V 2 i ) f y t dt + Formally use Itô product rule: d(w i W j ) = W i dw j + W j dw i + δ ij dt (prove using Itô integral definition) d i=1 V i f y t dw i t
Related PDE Consider Itô SDE for f y t (x) with initial data y (x) x: ( t f y t (x) = f x + Ṽ + d t + i=1 d i=1 Feynman Kac solution u(t, x) of PDE t u + ( Ṽ + d i=1 V 2 i is u(t, x) = E ( f (y t (x)) ) ) u = V 2 i ) V i f y τ (x)dw i τ f y τ (x)dτ with u(, x) = f (x) (Roughly, take the expectation of Itô SDE for f )
Black Scholes Merton PDE Good explanation Evans notes, p. 114 Constant volatility v, stock/index value u t evolves: du t = µu t dt + v u t dw t Current price of option at time t is C(t) = f (t, u t ) Itô formula and financial argument to duplicate C by a portfolio consisting of investment of u and a bond (risk-free with interest rate r) t f + ru u f + 1 2 vu2 uu f r f =
Stratonovich form (hereafter) Itô: T W i τ dw i τ = 1 2( W i T ) 2 1 2 T Stratonovich: T W i τ dw i τ = 1 2 ( W i T ) 2 Itô to Stratonovich: V = Ṽ 1 2 d i=1 V 2 i Stratonovich calculus familiar, easier, swapping to/back to Itô form trivial
Quadrature error I With τ q = t n + q t, q =,...,Q 1, Q t = h J 12 (t n, t n+1 ) = = = = tn+1 τ t n t n Q 1 τq+1 q= Q 1 τq+1 q= Q 1 q= dw 1 τ 1 dw 2 τ τ q W 1 τ W 1 t n dw 2 τ τ q (W 1 τ W 1 τ q ) + (W 1 τ q W 1 t n )dw 2 τ Q 1 J 12 (τ q, τ q+1 ) + q= ( W 1 τq W 1 t n ) W 2 (τ q )
Quadrature error II With τ q = t n + q t, q =,...,Q 1, Q t = h J 12 (t n, t n+1 ) Ĵ 12 (t n, t n+1 ) 2 = L 2 = Q 1 J 12 (τ q, τ q+1 ) 2 L 2 q= Q 1 ( t) 2 q= = Q( t) 2 = h 2 /Q J 12 (t n, t n+1 ) Ĵ 12 (t n, t n+1 ) = h/ Q L2 Wiktorsson improves to h/q (SDELab: Gilsing & Shardlow)